For pressure-driven, steady, fully developed laminar flow of an incompressible fluid through a straight channel of length L, we can define the hydraulic resistance as Rhyd = Δp/Q, where Δp is the pressure drop and Q is the flow rate (analogous to the electrical resistance Relec = ΔV/I, where ΔV is the electrical potential drop and I is the electric current). The following table summarizes the hydraulic resistance of channels with different cross sectional shapes [30]:
Calculate the hydraulic resistance of a straight channel with the listed cross-sectional shapes using the following parameters for water flow: μ = 1 mPa_s, L = 10 mm,a = 100 μm, b = 33 μm, h = 100μm, and w = 300 μm. Based on the calculated hydraulic resistance, which shape is the most energy efficient to pump water?
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